How to animate a growing plant with branches?
You'll need to do some manual tweaking and timing, no doubt. Automating this would make sense if you had a massive amount of plants to process.

I've used the following method when animating growing plants with branches which I've found relatively easy to do. Admittedly it was on 3dsMax, but I believe the same steps applies to Blender.

First off, I always work with a mesh. Then I build a quick rig/armature loosely following the branching of the plant. Usually automatic skinning works fine.

To animate it grow, I take the starting pose and scale each bone to zero. I flip the starting and ending keyframes so the plant grows instead of shrinks. And since every bone inherits the scale of its parent, you get a nice, gradually cascading effect.

Things to tweak at this point are the timing of when the branches start to grow (wait for the parent bone to grow to full size before beginning its own scaling, for example), and slight rotation of each branch.